## A Study of Certain Functional Equations for the [theta]-functions |

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absolute value affine transformation analytic function analytic solutions ar-plane arbitrarily arbitrary factor argument choice of root circle interval common factor complex plane conjugate imaginaries consequence continuous 1-1 correspondence continuous function denote dense set determined doubly periodic system entire function entire plane equivalent solutions everywhere vanishes exceptional values existence factor of form form 11 formulas fourth root func functional equation gives Hence hypothesis I-III inequalities infinity Jacobian limit 59 line of zeros log q negative non-analytic solutions normalization origin pairs of functions pi(x positive primitive root quotient R(x R-plane real axis real solutions realized replaced by x'/2 resulting equation right-hand member root of p(x root of unity satisfy segment set of equidistant set of points set of zeros simply or doubly simply periodic set simultaneously substitution Suppose system of zeros t?-functions tf(x three allied solutions three initial points tions uniquely vanish at a/2 vanishes identically vicinity

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Page 10 - The different kinds of solutions can be differentiated from one another by the manner in which the roots are distributed.